A Logarithmic Endpoint Refinement for Type-II Navier–Stokes Blow-up Scalings

Abstract

We record an endpoint refinement for Euler-type rescalings arising in potential Type-II blow-up scenarios for the three-dimensional Navier-Stokes equations. If the scaling function satisfies f(lambda)^2/lambda -> 0, the local kinetic energy of every fixed-scale rescaled slice vanishes, precluding a non-trivial Euler profile. Applied to f(lambda) = lambda^(alpha-1) / [log(e/lambda)]^gamma, this excludes the logarithmically damped endpoint alpha = 3/2, gamma > 0. At the pure critical scaling alpha = 3/2, gamma = 0, we also record a conditional link between a non-trivial local blow-up profile and atomic concentration of the terminal kinetic-energy measure. These results restrict potential Type-II blow-up scenarios and do not resolve global regularity.